An application of the inequality for modified Poisson kernel
- Gaixian Xue^{1} and
- Junfei Wang^{2}Email author
https://doi.org/10.1186/s13660-016-0959-6
© Xue and Wang 2016
Received: 14 September 2015
Accepted: 4 January 2016
Published: 22 January 2016
Abstract
As an application of an inequality for modified Poisson kernel obtained by Qiao and Deng (Bull. Malays. Math. Sci. Soc. (2) 36(2):511-523, 2013), we give the generalized solution of the Dirichlet problem with arbitrary growth data.
Keywords
growth property Dirichlet problem modified Poisson kernel1 Introduction and results
Let \(\mathbf{R}^{n}\) (\(n\geq2\)) be the n-dimensional Euclidean space. The boundary and the closure of a set E in \(\mathbf{R}^{n}\) are denoted by ∂E and E̅, respectively. The Euclidean distance of two points P and Q in \(\mathbf{R}^{n}\) is denoted by \(\vert P-Q\vert \). Especially, \(\vert P\vert \) denotes the distance of two points P and O in \(\mathbf{R}^{n}\), where O is the origin in \(\mathbf{R}^{n}\).
We introduce a system of spherical coordinates \((r,\Theta)\), \(\Theta=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})\), in \(\mathbf{R}^{n}\) which are related to Cartesian coordinates \((x_{1},x_{2},\ldots,x_{n-1},x_{n})\) by \(x_{n}=r\cos\theta_{1}\).
Let \(B(P,r)\) denote the open ball with center at P and radius r (>0) in \(\mathbf{R}^{n}\). The unit sphere and the upper half unit sphere in \(\mathbf{R}^{n} \) are denoted by \(\mathbf{S}^{n-1}\) and \(\mathbf{S}_{+}^{n-1}\), respectively. The surface area \(2\pi^{n/2}\{\Gamma(n/2)\}^{-1}\) of \(\mathbf{S}^{n-1}\) is denoted \(w_{n}\). Let \(\Omega\subset\mathbf{S}^{n-1}\), a point \((1,\Theta)\) and the set \(\{\Theta; (1,\Theta)\in\Omega\}\) are denoted Θ and Ω, respectively. For two sets \(\Lambda\subset\mathbf{R}_{+}\) and \(\Omega\subset\mathbf{S}^{n-1}\), we denote \(\Lambda\times\Omega=\{ (r,\Theta)\in\mathbf{R}^{n}; r\in\Lambda,(1,\Theta)\in\Omega\}\), where \(\mathbf{R}_{+}\) is the set of all positive real numbers.
For the set \(\Omega\subset\mathbf{S}^{n-1}\), we denote the set \(\mathbf{ R}_{+}\times\Omega\) in \(\mathbf{R}^{n}\) by \(C_{n}(\Omega)\), which is called a cone. For the set \(I\subset \mathbf{R}\), the sets \(I\times\Omega\) and \(I\times\partial{\Omega}\) are denoted \(C_{n}(\Omega;I)\) and \(S_{n}(\Omega;I)\), respectively, where R is the set of all real numbers. Especially, the set \(S_{n}(\Omega; \mathbf{R}_{+})\) is denoted \(S_{n}(\Omega)\).
Recently, Qiao and Deng (cf. [1]) gave the solution of the Dirichlet problem in a cone. Applications of modified Poisson kernel with respect to the Schrödinger operator, we refer the reader to the papers by Huang and Ychussie (see [6]) and Li and Ychussie (see [7]).
Theorem A
Furthermore, Qiao and Deng (cf. [4]) supplemented the above result and proved the following.
Theorem B
As an application of the inequality (1) and the generalized Poisson kernel \(PI_{\Omega}^{\phi}(P,Q)\), we have the following.
Theorem
2 Lemmas
Lemma 1
Proof
Lemma 2
(See [4])
Suppose that the following two conditions are satisfied:
3 Proof of Theorem
To see that \(H_{\Omega}^{\phi_{g}}(P)\) is a harmonic function in \(C_{n}(\Omega)\), we remark that \(H_{\Omega}^{\phi_{g}}(P)\) satisfies the locally mean-valued property by Fubini’s theorem.
For any \(\epsilon>0\) and a positive number δ, by (9) we can choose a number R (\(>\max\{1,2(t'+\delta)\}\)) such that (6) holds, where \(P\in C_{n}(\Omega)\cap B(Q',\delta)\).
Thus we complete the proof of Theorem.
Declarations
Acknowledgements
The authors are very thankful to the anonymous referees for their valuable comments and constructive suggestions.
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Authors’ Affiliations
References
- Qiao, L, Deng, GT: Growth property and integral representation of harmonic functions in a cone. Bull. Malays. Math. Soc. 36(2), 511-523 (2013) MathSciNetMATHGoogle Scholar
- Rosenblum, G, Solomyak, M, Shubin, M: Spectral Theory of Differential Operators. VINITI, Moscow (1989) Google Scholar
- Qiao, L: Integral representations for harmonic functions of infinite order in a cone. Results Math. 61(1-2), 63-74 (2012) View ArticleMathSciNetMATHGoogle Scholar
- Qiao, L: Growth of certain harmonic functions in an n-dimensional cone. Front. Math. China 8(4), 891-905 (2013) View ArticleMathSciNetMATHGoogle Scholar
- Qiao, L: Modified Poisson integral and Green potential on a half-space. Abstr. Appl. Anal., 2012, Article ID 765965 (2012) Google Scholar
- Huang, J, Ychussie, B: The modification of Poisson-Sch integral on cones and its applications. Filomat (to appear) Google Scholar
- Li, Z, Ychussie, B: Sharp geometrical properties of a-rarefied sets via fixed point index for the Schrödinger operator equations. Fixed Point Theory Appl. 2015, 89 (2015) View ArticleMathSciNetGoogle Scholar